Precalculus Examples

$4^{3 x - 7} = 32^{2 x}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$2^{2 (3 x - 7)} = 2^{5 (2 x)}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$2 (3 x - 7) = 5 (2 x)$

Step 3

Solve for $x$ .

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Step 3.1

Simplify $2 (3 x - 7)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (3 x - 7) = 5 \cdot (2 x)$

Step 3.1.2

Simplify by adding zeros.

$2 (3 x - 7) = 5 \cdot (2 x)$

Step 3.1.3

Apply the distributive property.

$2 (3 x) + 2 \cdot - 7 = 5 \cdot (2 x)$

Step 3.1.4

Multiply.

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Step 3.1.4.1

Multiply $3$ by $2$ .

$6 x + 2 \cdot - 7 = 5 \cdot (2 x)$

Step 3.1.4.2

Multiply $2$ by $- 7$ .

$6 x - 14 = 5 \cdot (2 x)$

Step 3.2

Multiply $2$ by $5$ .

$6 x - 14 = 10 x$

Step 3.3

Move all terms containing $x$ to the left side of the equation.

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Step 3.3.1

Subtract $10 x$ from both sides of the equation.

$6 x - 14 - 10 x = 0$

Step 3.3.2

Subtract $10 x$ from $6 x$ .

$- 4 x - 14 = 0$

Step 3.4

Add $14$ to both sides of the equation.

$- 4 x = 14$

Step 3.5

Divide each term in $- 4 x = 14$ by $- 4$ and simplify.

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Step 3.5.1

Divide each term in $- 4 x = 14$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{14}{- 4}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $- 4$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{14}{- 4}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{- 4}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Cancel the common factor of $14$ and $- 4$ .

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Step 3.5.3.1.1

Factor $2$ out of $14$ .

$x = \frac{2 (7)}{- 4}$

Step 3.5.3.1.2

Cancel the common factors.

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Step 3.5.3.1.2.1

Factor $2$ out of $- 4$ .

$x = \frac{2 \cdot 7}{2 \cdot - 2}$

Step 3.5.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 7}{2 \cdot - 2}$

Step 3.5.3.1.2.3

Rewrite the expression.

$x = \frac{7}{- 2}$

Step 3.5.3.2

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7}{2}$

Decimal Form:

$x = - 3.5$

Mixed Number Form:

$x = - 3 \frac{1}{2}$